76 research outputs found
Why Philosophers Should Care About Computational Complexity
One might think that, once we know something is computable, how efficiently
it can be computed is a practical question with little further philosophical
importance. In this essay, I offer a detailed case that one would be wrong. In
particular, I argue that computational complexity theory---the field that
studies the resources (such as time, space, and randomness) needed to solve
computational problems---leads to new perspectives on the nature of
mathematical knowledge, the strong AI debate, computationalism, the problem of
logical omniscience, Hume's problem of induction, Goodman's grue riddle, the
foundations of quantum mechanics, economic rationality, closed timelike curves,
and several other topics of philosophical interest. I end by discussing aspects
of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and
beyond," MIT Press, 2012. Some minor clarifications and corrections; new
references adde
A Simpler Proof of PH C BP[ÓØP]
We simplify the proof by S. Toda [Tod89] that the polynomial hierarchy PH is contained in BP[ÓØP]. Our methods bypass the technical quantifier interchange lemmas in the original proof, and clarify the counting principles on which the result depends. We also show that relative to a random oracle R, PHR is strictly contained in ÓØPR
Locality and Complexity in Path Search
The path search problem considers a simple model of communication networks as channel graphs: directed acyclic graphs with a single source and sink. We consider each vertex to represent a switching point, and each edge a single communication line. Under a probabilistic model where each edge may independently be free (available for use) or blocked (already in use) with some constant probability, we seek to efficiently search the graph: examine (on average) as few edges as possible before determining if a path of free edges exists from source to sink. We consider the difficulty of searching various graphs under different search models, and examine the computational complexity of calculating the search cost of arbitrary graphs
Inseparability and Strong Hypotheses for Disjoint NP Pairs
This paper investigates the existence of inseparable disjoint pairs of NP
languages and related strong hypotheses in computational complexity. Our main
theorem says that, if NP does not have measure 0 in EXP, then there exist
disjoint pairs of NP languages that are P-inseparable, in fact
TIME(2^(n^k))-inseparable. We also relate these conditions to strong hypotheses
concerning randomness and genericity of disjoint pairs
Average Dependence and Random Oracles (Preliminary Report)
This paper is a technical investigation of issues in computational complexity theory relative to a random oracle. We introduce āaverage dependence,ā an alternative method to Bennett and Gillās āmeasure preserving map technique and illustrate our technique by the following results
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